Theorem ltexprlemloc 7415

Description: Our constructed difference is located. Lemma for ltexpri 7421. (Contributed by Jim Kingdon, 17-Dec-2019.)

Hypothesis

Ref	Expression
ltexprlem.1	|- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.

Assertion

Ref	Expression
ltexprlemloc	|- ( A <P B -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) )

Distinct variable groups:   x, y, q, r, A   x, B, y, q, r   x, C, y, q, r

Proof of Theorem ltexprlemloc

Dummy variables z w f g h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltexnqi 7217	. . . . . 6 |- ( q <Q r -> E. w e. Q. ( q +Q w ) = r )
2	1	adantl 275	. . . . 5 |- ( ( A <P B /\ q <Q r ) -> E. w e. Q. ( q +Q w ) = r )
3		ltrelpr 7313	. . . . . . . . . 10 |- <P C_ ( P. X. P. )
4	3	brel 4591	. . . . . . . . 9 |- ( A <P B -> ( A e. P. /\ B e. P. ) )
5	4	simpld 111	. . . . . . . 8 |- ( A <P B -> A e. P. )
6		prop 7283	. . . . . . . . 9 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
7		prarloc 7311	. . . . . . . . 9 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ w e. Q. ) -> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) )
8	6, 7	sylan 281	. . . . . . . 8 |- ( ( A e. P. /\ w e. Q. ) -> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) )
9	5, 8	sylan 281	. . . . . . 7 |- ( ( A <P B /\ w e. Q. ) -> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) )
10	9	ad2ant2r 500	. . . . . 6 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) )
11	4	simprd 113	. . . . . . . . . . . . . 14 |- ( A <P B -> B e. P. )
12	11	ad2antrr 479	. . . . . . . . . . . . 13 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> B e. P. )
13	12	ad2antrr 479	. . . . . . . . . . . 12 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) /\ y <Q ( z +Q w ) ) -> B e. P. )
14		ltanqg 7208	. . . . . . . . . . . . . . . 16 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
15	14	adantl 275	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
16		elprnqu 7290	. . . . . . . . . . . . . . . . . . 19 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ y e. ( 2nd ` A ) ) -> y e. Q. )
17	6, 16	sylan 281	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. P. /\ y e. ( 2nd ` A ) ) -> y e. Q. )
18	5, 17	sylan 281	. . . . . . . . . . . . . . . . 17 |- ( ( A <P B /\ y e. ( 2nd ` A ) ) -> y e. Q. )
19	18	adantlr 468	. . . . . . . . . . . . . . . 16 |- ( ( ( A <P B /\ q <Q r ) /\ y e. ( 2nd ` A ) ) -> y e. Q. )
20	19	ad2ant2rl 502	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> y e. Q. )
21		elprnql 7289	. . . . . . . . . . . . . . . . . . . 20 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ z e. ( 1st ` A ) ) -> z e. Q. )
22	6, 21	sylan 281	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. P. /\ z e. ( 1st ` A ) ) -> z e. Q. )
23	5, 22	sylan 281	. . . . . . . . . . . . . . . . . 18 |- ( ( A <P B /\ z e. ( 1st ` A ) ) -> z e. Q. )
24	23	adantlr 468	. . . . . . . . . . . . . . . . 17 |- ( ( ( A <P B /\ q <Q r ) /\ z e. ( 1st ` A ) ) -> z e. Q. )
25	24	ad2ant2r 500	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> z e. Q. )
26		simplrl 524	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> w e. Q. )
27		addclnq 7183	. . . . . . . . . . . . . . . 16 |- ( ( z e. Q. /\ w e. Q. ) -> ( z +Q w ) e. Q. )
28	25, 26, 27	syl2anc 408	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( z +Q w ) e. Q. )
29		ltrelnq 7173	. . . . . . . . . . . . . . . . . . 19 |- <Q C_ ( Q. X. Q. )
30	29	brel 4591	. . . . . . . . . . . . . . . . . 18 |- ( q <Q r -> ( q e. Q. /\ r e. Q. ) )
31	30	simpld 111	. . . . . . . . . . . . . . . . 17 |- ( q <Q r -> q e. Q. )
32	31	adantl 275	. . . . . . . . . . . . . . . 16 |- ( ( A <P B /\ q <Q r ) -> q e. Q. )
33	32	ad2antrr 479	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> q e. Q. )
34		addcomnqg 7189	. . . . . . . . . . . . . . . 16 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
35	34	adantl 275	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) /\ ( f e. Q. /\ g e. Q. ) ) -> ( f +Q g ) = ( g +Q f ) )
36	15, 20, 28, 33, 35	caovord2d 5940	. . . . . . . . . . . . . 14 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( y <Q ( z +Q w ) <-> ( y +Q q ) <Q ( ( z +Q w ) +Q q ) ) )
37		addassnqg 7190	. . . . . . . . . . . . . . . . 17 |- ( ( z e. Q. /\ w e. Q. /\ q e. Q. ) -> ( ( z +Q w ) +Q q ) = ( z +Q ( w +Q q ) ) )
38	25, 26, 33, 37	syl3anc 1216	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( ( z +Q w ) +Q q ) = ( z +Q ( w +Q q ) ) )
39		addcomnqg 7189	. . . . . . . . . . . . . . . . . 18 |- ( ( w e. Q. /\ q e. Q. ) -> ( w +Q q ) = ( q +Q w ) )
40	26, 33, 39	syl2anc 408	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( w +Q q ) = ( q +Q w ) )
41	40	oveq2d 5790	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( z +Q ( w +Q q ) ) = ( z +Q ( q +Q w ) ) )
42		simplrr 525	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( q +Q w ) = r )
43	42	oveq2d 5790	. . . . . . . . . . . . . . . 16 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( z +Q ( q +Q w ) ) = ( z +Q r ) )
44	38, 41, 43	3eqtrd 2176	. . . . . . . . . . . . . . 15 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( ( z +Q w ) +Q q ) = ( z +Q r ) )
45	44	breq2d 3941	. . . . . . . . . . . . . 14 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( ( y +Q q ) <Q ( ( z +Q w ) +Q q ) <-> ( y +Q q ) <Q ( z +Q r ) ) )
46	36, 45	bitrd 187	. . . . . . . . . . . . 13 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( y <Q ( z +Q w ) <-> ( y +Q q ) <Q ( z +Q r ) ) )
47	46	biimpa 294	. . . . . . . . . . . 12 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) /\ y <Q ( z +Q w ) ) -> ( y +Q q ) <Q ( z +Q r ) )
48		prop 7283	. . . . . . . . . . . . 13 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
49		prloc 7299	. . . . . . . . . . . . 13 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ ( y +Q q ) <Q ( z +Q r ) ) -> ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) )
50	48, 49	sylan 281	. . . . . . . . . . . 12 |- ( ( B e. P. /\ ( y +Q q ) <Q ( z +Q r ) ) -> ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) )
51	13, 47, 50	syl2anc 408	. . . . . . . . . . 11 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) /\ y <Q ( z +Q w ) ) -> ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) )
52	51	ex 114	. . . . . . . . . 10 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ ( z e. ( 1st ` A ) /\ y e. ( 2nd ` A ) ) ) -> ( y <Q ( z +Q w ) -> ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
53	52	anassrs 397	. . . . . . . . 9 |- ( ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ z e. ( 1st ` A ) ) /\ y e. ( 2nd ` A ) ) -> ( y <Q ( z +Q w ) -> ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
54	53	reximdva 2534	. . . . . . . 8 |- ( ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) /\ z e. ( 1st ` A ) ) -> ( E. y e. ( 2nd ` A ) y <Q ( z +Q w ) -> E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
55	54	reximdva 2534	. . . . . . 7 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> ( E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) -> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
56		prml 7285	. . . . . . . . . . . 12 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. z e. Q. z e. ( 1st ` A ) )
57		rexex 2479	. . . . . . . . . . . 12 |- ( E. z e. Q. z e. ( 1st ` A ) -> E. z z e. ( 1st ` A ) )
58	6, 56, 57	3syl 17	. . . . . . . . . . 11 |- ( A e. P. -> E. z z e. ( 1st ` A ) )
59		r19.45mv 3456	. . . . . . . . . . 11 |- ( E. z z e. ( 1st ` A ) -> ( E. z e. ( 1st ` A ) ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
60	5, 58, 59	3syl 17	. . . . . . . . . 10 |- ( A <P B -> ( E. z e. ( 1st ` A ) ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
61	60	adantr 274	. . . . . . . . 9 |- ( ( A <P B /\ q <Q r ) -> ( E. z e. ( 1st ` A ) ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
62		prmu 7286	. . . . . . . . . . . . 13 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. x e. Q. x e. ( 2nd ` A ) )
63		rexex 2479	. . . . . . . . . . . . 13 |- ( E. x e. Q. x e. ( 2nd ` A ) -> E. x x e. ( 2nd ` A ) )
64	6, 62, 63	3syl 17	. . . . . . . . . . . 12 |- ( A e. P. -> E. x x e. ( 2nd ` A ) )
65		r19.9rmv 3454	. . . . . . . . . . . . . 14 |- ( E. x x e. ( 2nd ` A ) -> ( ( z +Q r ) e. ( 2nd ` B ) <-> E. y e. ( 2nd ` A ) ( z +Q r ) e. ( 2nd ` B ) ) )
66	65	orbi2d 779	. . . . . . . . . . . . 13 |- ( E. x x e. ( 2nd ` A ) -> ( ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. y e. ( 2nd ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
67		r19.43 2589	. . . . . . . . . . . . 13 |- ( E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. y e. ( 2nd ` A ) ( z +Q r ) e. ( 2nd ` B ) ) )
68	66, 67	syl6rbbr 198	. . . . . . . . . . . 12 |- ( E. x x e. ( 2nd ` A ) -> ( E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
69	5, 64, 68	3syl 17	. . . . . . . . . . 11 |- ( A <P B -> ( E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
70	69	rexbidv 2438	. . . . . . . . . 10 |- ( A <P B -> ( E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> E. z e. ( 1st ` A ) ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
71	70	adantr 274	. . . . . . . . 9 |- ( ( A <P B /\ q <Q r ) -> ( E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) <-> E. z e. ( 1st ` A ) ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
72		ibar 299	. . . . . . . . . . . . . . 15 |- ( q e. Q. -> ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) )
73	72	adantr 274	. . . . . . . . . . . . . 14 |- ( ( q e. Q. /\ r e. Q. ) -> ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) ) )
74		ibar 299	. . . . . . . . . . . . . . 15 |- ( r e. Q. -> ( E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) )
75	74	adantl 275	. . . . . . . . . . . . . 14 |- ( ( q e. Q. /\ r e. Q. ) -> ( E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) <-> ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) )
76	73, 75	orbi12d 782	. . . . . . . . . . . . 13 |- ( ( q e. Q. /\ r e. Q. ) -> ( ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) \/ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) <-> ( ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) \/ ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) ) )
77	30, 76	syl 14	. . . . . . . . . . . 12 |- ( q <Q r -> ( ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) \/ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) <-> ( ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) \/ ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) ) )
78		ltexprlem.1	. . . . . . . . . . . . . 14 |- C = <. { x e. Q. | E. y ( y e. ( 2nd ` A ) /\ ( y +Q x ) e. ( 1st ` B ) ) } , { x e. Q. | E. y ( y e. ( 1st ` A ) /\ ( y +Q x ) e. ( 2nd ` B ) ) } >.
79	78	ltexprlemell 7406	. . . . . . . . . . . . 13 |- ( q e. ( 1st ` C ) <-> ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) )
80	78	ltexprlemelu 7407	. . . . . . . . . . . . . 14 |- ( r e. ( 2nd ` C ) <-> ( r e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) )
81		eleq1 2202	. . . . . . . . . . . . . . . . 17 |- ( y = z -> ( y e. ( 1st ` A ) <-> z e. ( 1st ` A ) ) )
82		oveq1 5781	. . . . . . . . . . . . . . . . . 18 |- ( y = z -> ( y +Q r ) = ( z +Q r ) )
83	82	eleq1d 2208	. . . . . . . . . . . . . . . . 17 |- ( y = z -> ( ( y +Q r ) e. ( 2nd ` B ) <-> ( z +Q r ) e. ( 2nd ` B ) ) )
84	81, 83	anbi12d 464	. . . . . . . . . . . . . . . 16 |- ( y = z -> ( ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) <-> ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) )
85	84	cbvexv 1890	. . . . . . . . . . . . . . 15 |- ( E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) <-> E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) )
86	85	anbi2i 452	. . . . . . . . . . . . . 14 |- ( ( r e. Q. /\ E. y ( y e. ( 1st ` A ) /\ ( y +Q r ) e. ( 2nd ` B ) ) ) <-> ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) )
87	80, 86	bitri 183	. . . . . . . . . . . . 13 |- ( r e. ( 2nd ` C ) <-> ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) )
88	79, 87	orbi12i 753	. . . . . . . . . . . 12 |- ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> ( ( q e. Q. /\ E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) ) \/ ( r e. Q. /\ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) )
89	77, 88	syl6rbbr 198	. . . . . . . . . . 11 |- ( q <Q r -> ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) \/ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) ) )
90		df-rex 2422	. . . . . . . . . . . 12 |- ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) <-> E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) )
91		df-rex 2422	. . . . . . . . . . . 12 |- ( E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) <-> E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) )
92	90, 91	orbi12i 753	. . . . . . . . . . 11 |- ( ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) <-> ( E. y ( y e. ( 2nd ` A ) /\ ( y +Q q ) e. ( 1st ` B ) ) \/ E. z ( z e. ( 1st ` A ) /\ ( z +Q r ) e. ( 2nd ` B ) ) ) )
93	89, 92	syl6bbr 197	. . . . . . . . . 10 |- ( q <Q r -> ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
94	93	adantl 275	. . . . . . . . 9 |- ( ( A <P B /\ q <Q r ) -> ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> ( E. y e. ( 2nd ` A ) ( y +Q q ) e. ( 1st ` B ) \/ E. z e. ( 1st ` A ) ( z +Q r ) e. ( 2nd ` B ) ) ) )
95	61, 71, 94	3bitr4rd 220	. . . . . . . 8 |- ( ( A <P B /\ q <Q r ) -> ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
96	95	adantr 274	. . . . . . 7 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> ( ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) <-> E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) ( ( y +Q q ) e. ( 1st ` B ) \/ ( z +Q r ) e. ( 2nd ` B ) ) ) )
97	55, 96	sylibrd 168	. . . . . 6 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> ( E. z e. ( 1st ` A ) E. y e. ( 2nd ` A ) y <Q ( z +Q w ) -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) )
98	10, 97	mpd 13	. . . . 5 |- ( ( ( A <P B /\ q <Q r ) /\ ( w e. Q. /\ ( q +Q w ) = r ) ) -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) )
99	2, 98	rexlimddv 2554	. . . 4 |- ( ( A <P B /\ q <Q r ) -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) )
100	99	ex 114	. . 3 |- ( A <P B -> ( q <Q r -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) )
101	100	ralrimivw 2506	. 2 |- ( A <P B -> A. r e. Q. ( q <Q r -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) )
102	101	ralrimivw 2506	1 |- ( A <P B -> A. q e. Q. A. r e. Q. ( q <Q r -> ( q e. ( 1st ` C ) \/ r e. ( 2nd ` C ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 697   /\ w3a 962   = wceq 1331   E.wex 1468   e. wcel 1480   A.wral 2416   E.wrex 2417   {crab 2420   <.cop 3530   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   1stc1st 6036   2ndc2nd 6037   Q.cnq 7088   +Q cplq 7090   <Q cltq 7093   P.cnp 7099   <P cltp 7103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-enq0 7232  df-nq0 7233  df-0nq0 7234  df-plq0 7235  df-mq0 7236  df-inp 7274  df-iltp 7278

This theorem is referenced by:  ltexprlempr  7416